Solving Quadratic and Rational Inequalities

A quadratic inequality is an inequality that involves a variable term with a second-degree power. When solving quadratic inequalities, the rules of addition, subtraction, multiplication, and division of inequalities still hold, but the final step in the solution is different. Working out these quadratic inequalities is almost like a puzzle that falls neatly into place as you work on it. The best way to describe how to solve a quadratic inequality is to use an example and put the rules right in the example. A rational inequality involves a fraction with an attitude. You deal with the attitude using techniques similar to those used with quadratic inequalities.

When you choose a number to the left of –4, both factors are negative and the product is positive. Between –4 and 1, the first factor is negative and the second factor is positive, resulting in a negative product. To the right of 1, both factors are positive, giving you a positive product. Just testing one of the numbers in the interval tells you what will happen to all of them. Figure 15-5 shows you a number line with the critical numbers in their places and the signs in the intervals between the points.


Setting an inequality equal to 0 works fine as long as you can find numbers that work. When the expression has no critical numbers or solutions to setting it equal to 0, then the expression never changes sign. It’s always negative or always positive. You only have to determine whether anything solves the problem. For example, the expression x2 + 4 in the inequality x2 + 4 > 0 doesn’t factor. And any number you put in for x gives you a positive value on the left. So this statement is always positive, and the inequality is true for all numbers.

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